is imposed in the method (see [1]} for further details). By discretizing the streamwise derivatives using first-order implicit Euler differences, (REF ) can be marched in the downstream direction starting from a station close to the nozzle, following the usual PSE method. The method allows us to update the values of \(\mathbf {\hat{q}}\)  at each axial station by solving the resulting discretized linear system, and the wavenumber \(\alpha \)  using (REF ). This leads to the final solution of the PSE for each frequency and azimuthal wavenumber.
